Sub-nanometer mapping of strain-induced band structure variations in planar nanowire core-shell heterostructures

Strain relaxation mechanisms during epitaxial growth of core-shell nanostructures play a key role in determining their morphologies, crystal structure and properties. To unveil those mechanisms, we perform atomic-scale aberration-corrected scanning transmission electron microscopy studies on planar core-shell ZnSe@ZnTe nanowires on α-Al2O3 substrates. The core morphology affects the shell structure involving plane bending and the formation of low-angle polar boundaries. The origin of this phenomenon and its consequences on the electronic band structure are discussed. We further use monochromated valence electron energy-loss spectroscopy to obtain spatially resolved band-gap maps of the heterostructure with sub-nanometer spatial resolution. A decrease in band-gap energy at highly strained core-shell interfacial regions is found, along with a switch from direct to indirect band-gap. These findings represent an advance in the sub-nanometer-scale understanding of the interplay between structure and electronic properties associated with highly mismatched semiconductor heterostructures, especially with those related to the planar growth of heterostructured nanowire networks.


Supplementary Note 1. Growth directions, plane interactions and associated mismatches
On C-plane (0001) sapphire the guided nanowires grow along six m ±⟨11 00⟩ directions, which reflect the three-fold symmetry of the plane and forming 60º between each growth direction ( Supplementary Figure 1a Lattice mismatch between substrate and nanowire is dependent on substrate orientation as the interacting planes are different in both cases. In addition, in order to take into account the plastic relaxations and evaluate the remaining mismatch we used ′ % 100

Supplementary
Supplementary Equation (2) where a misfit dislocation occurs every m substrate planes and n material planes. Based on the calculated epitaxial relationships and associated mismatches in Supplementary Table 2, ZnSe nanowires grown following ±[11 00] directions mainly relax strain by the creation of misfit dislocations. The residual strain is elastically accommodated within the first few nanometers of core growth, leading to a relaxed ZnSe lattice close to the junction.

Supplementary
7 Geometric Phase Analysis (GPA) applied to the (111) horizontal planes reveals a plane bending from -0.5º to +2º in A-oriented sapphire. However, the rotation inversion at both sides of the NW is gradual, so there is no appearance of a sharp boundary as in the purely cylindrical core presented in Figure 2  Top: HAADF micrograph, overall dilatation and overall rotation maps. Bottom: strain matrix components xx, yy and xy maps.

Supplementary Note 4. Atomic modelling on non-faceted cores 1 st Approximation
The absence of bending in C-substrate nanowires can be attributed to the faceting strictly parallel to (202 ) planes. The effective interplanar distance of shell planes presenting a plane bending interacting with the core is given by = / cos > Supplementary Equation (3). Considering that dshell > dcore, the effective mismatch in a sharp interface increases with an increasing angle .
Supplementary Figure 6. Schematics of interplanar distances and angles when plane bending occurs.
The origin of plane bending in A-plane sapphire is related to the atomic steps originating from the curvature in nanosized geometry. For this reason, the closer to flat (202 ) surfaces, the lower bending angles minimize elastic energy accumulation.

STEM Simulations
Atomic models of the top left section of the nanowires have been created for relaxed core and shell lattices but inducing a crystal cell rotation around the nanowire growth axis ranging from 0º to 5º rotation, with steps of 1º. For each of them, the corresponding HAADF micrograph has been simulated reproducing the imaging conditions. Rotation maps of each micrograph have been obtained through GPA applied to (111) and (202 ) planes.

Supplementary Figure 7.
Plane rotation obtained through GPA applied to (111) and (202 ) planes applied to atomic models with shell plane rotations ranging from 0º to 5º in order to evaluate the number of dislocations and plane matching.
Rotation maps of Figure 3 in the main manuscript visually reveal a minimization of elastic energy at the interface for a situation in which the shell crystal is rotated by 2º, as measured experimentally. We can consider the number of dislocations created by lattice mismatch as a way to quantify lattice adaptation, since an excess of elastic energy leads to the formation of dislocations (Supplementary Table 3

Nanowire with 10 nm radius
The same analysis based on atomic modelling and HAADF STEM image simulations has been applied to a modelled nanowire with smaller radius (10 nm). When analysing the situation of a smaller core diameter (i. e. higher curvature), the induced rotation of 2º does not correspond to the most stable situation, which shifts to higher angles. Taking into account the number of dislocations, as listed in Supplementary Table 4, a minimum is reached in the case of 0º and 3º. The matching between planes tangent to the interface is smoother with the 3º bending (highlighted with a white arrow), while elastic energy is accumulated without the formation of a misfit dislocation for the cases of 0º, 2º and 4º rotations.
We conclude that a minimization of elastic energy is reached with plane rotations close to 3º for this core size. Therefore, plane bending for elastic energy minimization in the case of higher curvature shifts to higher angles.

Nanowire core with 35 nm radius
From the conclusions obtained after analysing situations presenting small cores, we can predict that bending would tend to zero in the case of bigger core diameters. For this reason, the same methodology is applied to a core presenting a 35 nm radius. Figure 9. Rotation maps on (111) and (202 ) planes obtained through Geometric Phase Analysis on an atomic model of the core-shell structure with 35 nm core radius.

Supplementary
As predicted, a better lattice accommodation is reached for lower bending angles. When analysing the total number of dislocations, as displayed in Supplementary Table 5, the minimal number of 20 dislocations is achieved with rotations ranging from 0 to 1º. Checking the matching of planes tangential to the core surface we find that 0º is far from optimal matching (black arrow), but the matching is improved for 1º bending in the case of (111) planes and 0.5º in the case of (202 ) planes. Elastic energy minimization is therefore assumed to happen between 0.5º and 1º. At the same time, we assume from this analysis that radii > 50 nm need to be achieved for equalling the flat plane situation of C-sapphire flat nanowires. The analysis performed on different core radius diameters (i.e. different curvature) confirms the hypothesis that the main cause of the anomalous grain boundaries are the atomic steps caused by the curvature of the core. From the results obtained after modelling greater nanowire cores, the core dimensions should be greater than 70 nm to fully avoid plane bending, which would lead at the same time to a decrease in surface / volume ratio.

Supplementary
Controlling the wetting in surfaces for reaching contact angles of 90º appears as a more promising candidate, as plane rotations are avoided even for reduced NW core sizes.

2nd approximation
A second, more descriptive geometrical model has been developed to predict the position and number of misfit dislocations at the core-shell interface. This second approximation considers the effect of the core radius, r, and the angle the shell planes have with respect to those in the core.
In this model, we compute the number of dislocations per shell (nd), given the radius of the core and the angle. Here we unveiled the details on how the non-linear variation of the effective distance between planes makes the dislocations arise, by computing the density of dislocations per length unit and validating the positions obtained by the atomistic simulations.

Supplementary Equation (4)
Where is the density of dislocations per perimeter unit.
Dislocations arise due to the different cell parameters, and consequently, different interplanar distances, in the substrates and in the materials grown on top. This scenario requires to define the mismatch differently, as no substrate is present, but a material next to another. Then, this mismatch ms can be defined as follows, for a material 1 and 2: Supplementary Equation ( First we set the distances and parameters involved in the system: -The ZnSe core (specified as Se in the equations, being aSe its cell parameter), with (hkl) = (111) planes perfectly horizontal and parallel to the substrate. Since no rotation is involved for these planes, each time we add a plane to the core we must just sum the interplanar distance, defined by: (6) -A different consideration applies to the ZnTe (specified as Te in the equations, aTe being its cell parameter). Intuitively, at the very bottom, where the circular core shape begins, no curvature is observed, with only a continuous plane appearing. Nevertheless, if we start going up in height, the effect of the curvature becomes noticeable, and this infinite vertical plane becomes a bent curve that heavily influences the disposition of the atoms.
Therefore, we define an effective distance for the ZnTe, deff, which is also related to its interplanar distance, dTe=aTe/√3.
Then, the condition that must be fulfilled for creating a dislocation becomes: , Supplementary Equation (7) where y is the height coordinate and we want to obtain the values for y at which the core and shell planes coincide. This can be done as both regions can be treated as mathematically independent from each other and just defined by the shape of their boundary. Of course, they are physically related, as the core defines how the shell grows, but we must think as if we could separate them both once having grown.
The definition of the increments in height defined in the core, due to the perfectly parallel condition is then very simple: Supplementary Equation (8) Next, we geometrically define the shell planes simulating the horizontal planes rotated degrees, to eventually use them to get the effective distance in contact with the core. To do this parametrization, the system in Supplementary Figure 10a, is taken as a reference. To define the linear function that describe the first shell plane, we use the coordinate (r,0) as its origin.
In addition, as defined in the first approximation, the vertical projection of the planes is defined as p: Supplementary Equation (9) which is needed to compute the previously mentioned parametrization, as all the lines/planes of order m meet the condition of intersecting at (r, mp). The other parameter we need to fully characterize these lines is the slope, which is directly defined by the angle, : → Supplementary Equation (10) Then, for the first plane, order 0 (m=0): , 0 → 0 ; Supplementary Equation (11) 0: For the second plane, order 1 (m=1): , , → ; Supplementary Equation (12) 1: And, the m th order plane follows: Supplementary Equation (13) : Note that we have defined that the first shell plane perfectly intercepts with the first geometric coordinate of the core. However, the most complete case would be to add an extra parameter with a value between 0 and the deff between plane m=0 and m=1, that would account for a possible shift in the growth of this first plane. Anyway, this addition would only slightly shift the obtained positions for the dislocations thought the perimeter. Nonetheless, it is important to keep its influence in mind, as it is the explanation for why some obtained dislocations are slightly displaced with respect to those in the atomistic models, together with the apparition or not of dislocations at the very bottom or very top of the core-shell interface.
With the previous parametrization, we have already considered the influence of the rotation angle in the model. To consider the effects of the curvature of the core, we must go one step further. Interestingly, these effects are clearly visible in Supplementary Figure 10b. The steps that are being created while we go to the top of the core vary a lot locally, and this is what we compute in the next and final step.
The key point here is that we must compute the direct intersection of the previously parametrised planes with the core circumference itself, defined in Supplementary Figure 10b, as the coordinates c and e. Keeping in mind the coordinate system, we initiated in Supplementary Figure 10.a, and the coordinates in b) we can define: (14) Strictly speaking, to get c, we should set m=1, but we can develop the previous equation in order to get to the general formulation: Supplementary Equation (15) By keeping only the positive solution, as the negative belongs to the diagonally opposed quadrant, we can get the abscissa of the intersection with plane m, xm: And the corresponding ordinate, ym: Supplementary Equation (18) Summarising, the coordinates (xm,ym) indicating the intersection of the shell planes with the core circumference are computed, allowing us to calculate the effective increments the core sees from the shell.
In order to limit the values to study next, we can define the total number of planes in the core that will be needed to fill the entire core with (111) ZnSe planes: Supplementary Equation (19) Likewise, for the shell, the mmax fulfils the following condition, which can be numerically solved: Supplementary Equation (20) Eventually, what we must do is compute all the ordinates, for each plane of order between 0 and nmax for the core, and between 0 and mmax for the shell, and compare them to identify those y that satisfies the condition , , and those that maximise the difference between coincident ordinates, i.e., dislocation positions. To do so, we compute the y at which the intersection with the core shape happens, and we correlate them with the absolute value of the difference between the ym of the shell and the yn of the core = | ym -yn |. The first perfect coincidence happens at y=0, as we have defined it in the model (that is why it is important to consider the influence of the previously mentioned shift parameter). This means that if we subtract the planes of same order n = m, we will find a | ym -yn |=0 (i.e., intercept with the abscissa in absolute value functions), at y = 0. To find the next position of maximum coincidence, we must increase the order of the core planes compared to the shell, as the core has smaller projected distances in y. Then, when plotting y = | ym -yn+1 |, we will obtain a 0 at the position of maximum coincidence between planes, meaning that between this minimum and the one computed before with m=n, y=0, we must find a dislocation. In fact, as can be seen perfectly straight lines, in agreement with being equivalent to the first approximation, shows equally spaced positions for the perfect matches and mismatches. This is because the effective distance increase of this scenario is not affected at all by the curvature. In the 2º case, we see that the dislocation spacing is increasing as we go to the top of the core, and after the 3 rd dislocation, the spacing of the shell planes becomes stabilised (i.e., varies only slightly) and never reaches a | ym -yn |=0 condition again, meaning no new dislocation is generated. Since the formula we obtained is only affected by the angle and the curvature, we can be sure that this effect is only caused by the geometric effect that this exact rotation angle has on this exact core radius. This means, as we prove in the atomistic simulations with different radii, that the ideal rotation angle is solely defined by the core shape. On the other hand, with the 5º scenario, we observe an interesting effect caused by the excessive rotation compared to the ideal one. This extra rotation has the effect that the otherwise stabilised effective distance of the shell in the 2º case now affects too much the variation of the shell effective distance, making the inversion of the usual tendency of the shell plane of one order below the core plane to meet the next planar coincidence, but return to meet the previous one that was met before. This is visible in the atomistic simulation for the 5º case, in Supplementary Figure 10.f, in which the colour pattern of the displayed dislocations is inverted due to this behaviour. Therefore, while the 2º case started to show this inversion pattern but never resulted in an extra dislocation due to the geometrical constraints, the overrotation of this simulated 5º case led to a different kind of dislocation that we call dislocation inversion, making it a less favourable scenario.

Supplementary Note 5. Additional details on strain relaxation mechanisms and shell rotation
It is important to note that the observed shell behaviour relates to the nanometric size of the core. First, for perfectly parallel core and shell atomic planes merging at the interface, we would expect a comparatively higher effective mismatch between both materials. However, if the atomic planes at the shell suffer elastic deformation and bend then the effective plane spacing of ZnTe at the interface is increased (Supplementary Figure 6). In addition, the small size of the rounded core cross-section renders its surface atomically stepped, resulting in high crystallographic matching when the effective interplanar distance of ZnTe is modified by the rotation of the shell lattice with respect to the core. Our atomic models created for nanowires presenting 10 nm core radius (higher curvature) and 35 nm core radius (lower curvature) show that there is also a core radius dependence on the optimal shell plane rotation angle for elastic energy minimization (Supplementary Figure 10). Therefore, the size of the catalyst droplet used during the VLS growth, 1 as well as its contact angle with the considered surface, play a key role in defining the core morphology and enabling the shell lattice to adapt to it. To model this behaviour, we developed a purely geometrical model that incorporates invariant interplanar distances in both ZnSe and ZnTe and accounts for the circular shape (of arbitrary radius) of the core, allowing the shell planes to rotate any angle around it. By forcing the model to fit the experimental radius of the core and different shell rotations (i.e.: 0, 2 and 5º), it captured the interplay between the curvature of the core and the optimal rotation angle of the shell required to minimize plastic deformations (misfit dislocations). The model supports a ±2º optimal rotation, given a 20 nm radius, also in agreement with atomistic simulations by explaining how the effective interplanar distance of the shell at the interface with the core is slightly varied along the curvature to avoid extra misfit dislocations. Larger rotation angles (e.g.: 5º) induce a variation of this effective distance that is too high and leads to the so-called dislocation inversion phenomenon that generates additional plastic strain, as observed in Supplementary   Figure 4 "2nd Approximation".

Supplementary Note 6. Details on the Core-shell misfit dislocations
Our samples present two different types of misfit dislocations visible on our projected visualization axis:

1)
On the lateral sides of the NW core-shell interface, misfit dislocations consist of the addition of a full (111) plane in the core with respect to the shell.

2)
On the top side of the NW core-shell interface, misfit dislocations consist of the addition of half (10-1) plane in the core with respect to the shell.
Both types of defects are pure edge dislocations. In Supplementary Figures 11 and 12 we

Supplementary Note 7. Simulation of VEELS spectra
We have carried out STEM-VEELS simulations to study the influence of parasitic radiation contributing to the inelastic electron signal. Our simulations are based on classical electrodynamics, with the materials described by their frequency-dependent, local dielectric functions. The energy-loss probability is the sum of a bulk contribution, which is independent of geometry and proportional to the path travelled in each material, and a surface term determined by the interfaces and their distribution. 2  imaging. 12 We applied the described methods to compute a spectral image consisting of 84x60 pixels in which the surface terms are obtained by incorporating a geometry deduced from the experimental images. We then obtained the total bulk distribution by adding the bulk contribution with or without retardation, as obtained from the expressions above with the dielectric function corresponding to each material at the beam position (i.e., Γ Γ , Γ ).
In the represented system, the main influence on the surface term is due to the cylindrical shape of the core and the shell. These geometries are reliably simulated within the BEM calculations, using the vertical and horizontal dimensions of both the core and the shell. In addition, the sapphire substrate and the model platinum coating have, in order to keep a computationally realistic grid, a region similar in size to the one observed in the spectral images, but as a result, smaller than the actual configuration of the system. This thinner sapphire substrate induces the appearance of an intense waveguide mode peak centred at 1.17 eV that influences the signal that can be read in this energy region for the two main semiconductors. We proved that the emergence of this peak, of which there is no evidence in the experimental data, is due to the artificial addition of the extra interface (sapphire-air) when defining the sapphire substrate. Modification of the sapphire region dimensions revealed a variation in the position and width of this peak in the surface term of the loss probability, validating its origin. The presence or absence of this peak can be neglected as it is far away from the targeted energies in which signatures from electronic states should start to arise (>2 eV). Another feature that needs to be taken into account is the surface peak that appears in the In order to apply the data treatment routines explained below, the first step was to extract the spectrum-image to a format readable by Python3, the language in which the code has been developed. To do so, a DigitalMicrograph custom script is prepared to apply a generalized power law background model to the full spectrum image as a reliable tool for removing the ZLP. Moreover, each pixel has been smoothed down by a Savitzky-Golay filter with a second order polynomial, and the resulting data extracted as tabbed text. 13,14 The main plasmon peaks of the ZnTe (centered at 15.3 eV), ZnSe (centered at 16.9 eV) and Al2O3 (centered at 25.4 eV) are used to map the local distribution of these materials. These The following considerations have been taken into account: 1) Every pixel in the experimental spectrum-image has been labelled with the material in which it was acquired, thanks to the thresholded mapping of their corresponding plasmons.
2) Every pixel in the simulated spectrum-image has been labelled with it the corresponding material. In addition, each pixel has its own calculation for each of the following components: total signal (Γ ), surface term (Γ ), retarded bulk term (Γ ) and non-retarded bulk term (Γ ).
For every pixel in the simulated spectrum-image, the following parameter called "Correction ratio", δ ′, , is computed per pixel:

Supplementary Equation (25)
This ratio is always a value between 0 and 1, and it represents the amount of signal or loss probability that comes intrinsically from the material at a given energy, , and position of the simulated spectrum-image, ′, (in other words, signal free of all kinds of additional contributions), compared to the total signal or loss probability that also includes Cherenkov radiation and waveguide modes arising from the interfaces. Note that the correction ratio involves loss probabilities that depend linearly on the thickness of the considered device. More precisely, the surface term obtained in the simulation is a loss probability per unit length along the thickness of the sample. Upon multiplication by the lamella thickness, each of the three terms in the ratio has the same linear dependence on thickness, thus making the ratio independent of this variable and robust against sample-to-sample thickness variations. In the present instance, the thickness was determined for ZnSe and ZnTe to have a uniform distribution around 45 nm. This is corroborated by the smooth thickness regions observed in Supplementary Figure 16, where variations in colour scale between materials are attributed to differences in the material-dependent inelastic mean free path, and not thickness changes.
Moreover, the absolute thickness estimate, performed for the main materials ZnSe and ZnTe, revealed comparable average thicknesses of 44±3 nm and 46±1 nm, respectively. Furthermore, the indirect fitting performs significantly better than the direct one at the ZnTe interface pixels. As a control case, the same comparison of fitting coefficients is done with the bulk ZnTe pixels. Fitting an indirect gap in these pixels appears to result in a better goodness of fit as well, compared to a direct gap fit. However, in this case, the increase in the r 2 parameter can be solely attributed to an unphysical fit of the indirect curve to the tails of the waveguide mode clearly visible around 2.3eV, which give a much smaller and more disperse (2.1±0.3eV) result than the direct one. All the r 2 values, and the corresponding errors of the measurement, can be consulted at the final part of this supplementary note, "Significance tests". The worse performance of the direct curve fit in pure statistical terms (i.e., r 2 ) can be explained by the strong effect that the aforementioned waveguide mode and the pre-correction basal bulk Cherenkov radiation have around these energies, which produce an uneven curve making the fitting difficult. It is important to mention that due to waveguide modes being peaks, the sources of differences between the experimental system and the simulated one (i.e., slight geometry changes, assumed translational invariance and local thickness variations, delocalisation…) make these modes the most challenging to identify. Hence the residuals of the tails of the most intense and shifted peaks can remain in the final curves, although most of their contribution is erased by the corrections (Supplementary Figure 18). For instance, the convolution of the tails of the waveguide mode centred around 2.2 eV, visible in the ZnTe domain as in Supplementary   Figures 18c and d,  Supplementary Figure 18 presents some of the most common scenarios encountered regarding the global performance of the method. The presence of local negative signal (counts) can be spotted in some individual, but only anecdotally and with no apparent localisation. Since the ZLP extraction follows the same power law model throughout the entire spectrum image, any artifacts causing a deviation from this general model in the corresponding spectral region can result in a misfitted ZLP at a local level (pixelwise). Nevertheless, by studying the full spectrum-image we can confirm that this is not a common issue. Only a 1.59% of the total pixels, randomly distributed throughout the spectrum-image, had a 30% of channels with negative counts between 2 and 3 eV and with the average of these negative counts being larger in absolute terms than the 30% of the average of the remaining positive counts. Even when changing from 30% to 20% of channels in both considerations, which is much less harmful for the fitting, the value only rises to 4.69%. Moreover, this issue would require special caution in the percentage corresponding to ZnTe pixels, as the negative counts are typically centred Consequently, we expect that any band-gap shift of ±1 eV with respect to tabulated values to be properly captured by the model. Indeed, this range would be enough to resolve most of the phenomena that can induce band shifts. For larger band-gap shifts, it would be difficult to ensure the validity of the method that keeps the reference value as the one tabulated. In order to generalise this method and adapt it to larger variations, simulations should also consider band modulation by strain (e.g., from ab initio calculations).

Significance tests:
The following statistical tests have been performed to prove the statistical veracity of the statements we make throughout the article, in particular when comparing band gap values between regions, or when comparing different fit models. The values tabulated for the t distribution, which can be treated as a Gaussian given the number of samples used, are onesided for the sake of our test purpose. 21 Test 1: ZnTe interface pixels. Indirect fit vs Direct fit Let us test if the indirect fit is better than the direct one in ZnTe interfacial pixels.

762.12
Supplementary Equation (31) The null hypothesis is rejected, and we can accept the alternative hypothesis with a confidence higher than 99.9995%: i.e., the indirect fit is significantly better than the direct one in ZnTe interface pixels. Let us test if the indirect fit is better than the direct one in ZnTe bulk pixels. The null hypothesis is rejected, and we can accept the alternative hypothesis with a confidence higher than 99.9995%: i.e., the indirect fit is significantly better than the direct one in ZnTe bulk pixels. The null hypothesis is rejected, and we can accept the alternative hypothesis with a confidence higher than 99.9995%: i.e., there is a significant decrease in the band gap energy of the ZnTe interface compared to the bulk ZnTe.

40.987
Supplementary Equation (34) The null hypothesis is rejected, and we can accept the alternative hypothesis with a confidence higher than 99.9995%: i.e., there is a significant decrease in the band gap energy of the ZnSe interface compared to the bulk ZnSe.